Polynomial irreducible tensors for point groups

Abstract

Generating functions for (Γ_r,Γ_m) tensors for any point group G and any pair of its irreducible representations Γ_r and Γ_m are calculated explicitly. A (Γ_r,Γ_m) tensor transforms according to Γ_r, and its components are polynomials in another tensor transforming by Γ_m. Explicit integrity bases of (Γ_r,Γ_m) tensors are given for all pairs Γ_r and Γ_m for the groups C_n, D_n, T, and O, and for the same groups with reflections. A composition rule for extending the result to reducible representations is formulated. Point group tensors irreducible with respect to SO(3) are obtained, together with their generating functions.